Going to the limit when the quantum reference frame considered
becomes classical has been well analyzed, for example in
Ref.[1]}. The latter article analyzes the (reference frame
transformation) symmetry picture in quantum physics rigorously
though only in terms of applying the symmetry to the physical
system, instead of formulating description of the system and the
theory about it from the language of representation theory as
mentioned above. It tackles in detail how practical considerations
of the symmetry transformations as applied between physical
frames of reference, with physical quantities defined accordingly
as relative to the frames, may realize the mathematical idealization
in which no reference frame is explicitly considered. The relevant
symmetry is naturally taken essentially as one of a Lie group. Of
course the results are qualitatively as one would naively expect.
When the reference frame can be well approximated as a classical
one, the idealized mathematical symmetry picture of the Lie group,
as described above, is well
realized. The symmetry transformation picture analyzed there
is not one of our changes in the noncommutative values but the
more conventional one of changes in the statistical distributions
of the projective measurement results. The noncommutative
values for the observables obviously reduce to the classical
real number values when the physical state descriptions of the
reference frames involved can be taken as essentially classical.
Moreover, it is interesting to note that the key mathematical
objects involved for the physical properties of an observable,
as on a state, in the analysis of Ref.[1]} is really the
expectation value function, from which the corresponding
statistical distribution result is to be retrieved. Since the
noncommutative value is really like a local representation
of the expectation value function, our noncommutative values
of changes picture is rather fully compatible with the analysis.
